The extension of (REF ) to a conformally flat spacetime \({\cal M}^3\)  is obtained [1]} by replacing \(\partial _a \rightarrow \nabla _a\) . Since the commutator (REF ) vanishes in every conformally flat spacetime, all the properties of \({\mathfrak {C}}_{\alpha (n)}(h)\)  follow through.
However, in addition to the Lorentz generators, \(\nabla _a\)  involves the special conformal generators \(K_b\)  (in a special gauge where the dilatation connection is gauged away).
In order to switch from the description in terms of \(\nabla _a\)  to that of the standard Lorentz covariant derivative \({\cal D}_a\) , one has to go through the process of degauging
[2]}, [1]}. This includes expanding \(\nabla _{a}\)  using the relation
\(\nabla _a=\mathcal {D}_a+\frac{1}{2} P_{a}{}^{b}K_b\) 
